The Pareto domination of irrational expectations over rational expectations

JOURNAL

OF ECONOMIC

THEORY

46,

322-334

(1988)

The Pareto Domination of Irrational Expectations over Rational Expectations MICHAEL TEIT NIELSEN* Institute

of Economics, University of Copenhagen, Copenhagen, Denmark, and Department of Economics, Princeton University, Princeton, New Jersey 08540

Received

March

25, 1987; revised

October

25, 1987

A simple overlapping generations economy with one consumer in every generation and with one consumption good in every period is considered. It is shown that a sequence of temporary equilibria generated by an “irrational” inelastic expectation function always generates an allocation which Pareto dominates the corresponding allocation generated by the rational expectation function. Furthermore, it is shown that in spite of systematic prediction errors, such an allocation is also efficient. Journal of Economic Literature Classification Numbers: 021, 024, ‘? 1988 Academic Press. Inc. 026.

1. INTRODUCTION In this paper we examine the welfare implications of different types of expectations. We use the standard one-consumer/one-consumption good overlapping generations (OLG) economy, where we compare sequences of temporary equilibria. Rational expectations, or perfect foresight, give the expectations that yield the maximal outcome (utility) for the individual, given that prices are what they are. Therefore it is tempting to infer that allocations generated by rational expectations must be Pareto optimal. It also seems obvious that allocations generated by inconsistent (“irrational”) expectations must have very poor efficiency properties. Nevertheless, in this paper we show that such allocations generated by inelastic expectation functions Pareto dominate the corresponding rational expectations allocations, and also are Pareto optimal. * I have benefitted from comments from colleagues at the University of Copenhagen, in particular Birgit Grodal, Hans Jorgen Jacobsen, and Christian Schultz. Ariel Rubinstein, while visiting Princeton University, also gave helpful comments. Financial support from the Danish Social Sciences Research Council, the Fulbright Commission, the Knud Hejgaard Foundation, and the Laurits Andersen Foundation, for studying one year at U. C. Berkeley. is gratefully acknowledged. 322 0022-0531/88

$3.00

Copyright 6, 1988 by Academic Press. Inc. All rights 01 reproducfmn in any lorm reserved.

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In Section 2 the simple overlapping generations economy is introduced, and we define the concepts to be used. In Sections 3 and 4 we analyze the dynamics of the OLG economy in the special cases of rational expectations and inelastic expectations. In Section 5 the relative efficiency of the two types of allocations is considered, and in Section 6 the absolute efficiency in the case of inelastic expectations is analyzed. Finally, some concluding remarks are made in Section 7. 2. THE MODEL We use the by now quite popular overlapping generations economy [ 121 with one identical consumer in each generation living for two subsequent periods, and with one consumption good in each period. Time is discrete, t = 0, 1, 2, ...) co, period 1 being the first that we wish to analyze, and period 0 being necessary for starting up the expectations. For a more detailed presentation of the OLG model, see [6], and of temporary equilibrium theory, see [7, 131. Let us explain verbally how the economy works: In each period there is an old agent who is endowed with the quantity e, of the consumption good and a positive stock of outside money, M, and a young agent who is endowed with e, of the consumption good. There is a market for the nonstorable consumption good. Money is taken to be numeraire (we consider only monetary equilibria), while the consumption good price is p, The natural interpretation of the economy is that it is a pure exchange economy, although the model can be interpreted as having some special cases of production activities [8]. The young agent can transfer value from youth to old age by selling some of the consumption good endowment and buying money which can be carried over to the subsequent period and sold, when this agent is old. Note that we do not allow the young agent to dissave. The decision of how much to save will depend on the agent’s expectation of the consumption good price of the subsequent period. In a temporary equilibrium the price (vector) that comes about in a given period is a price that clears the markets in that period, given the expectations of the young agent. Let us be a little more precise now. The consumer has as consumption set the strictly positive cone IR: +, and is endowed with a vector of the consumption good, e = (e,, e,), belonging to rW: + . (The first dimension is related to consumption in youth, the second in old age.) The preferences of the consumer can be represented by the utility function U. We assume: Assumption Al. UE C’([w: +, R), and U is strictly increasing and strictly concave, and each level surface of U has strictly positive Gaussian curvature and has its closure contained in RZ, + .

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MICHAEL TEIT NIELSEN

Assumption A2.

8 := D, U(e,, e,)/D, U(e,, e2) < 1.

Assumption Al is rather standard, assuming strict monotonicity and convexity of preferences, and smoothness such that utility maximization gives a smooth demand function. Assumption A2 is less innocent. It is an assumption assuring that we are in what Gale [6] calls “The Samuelson Case”: The youngster saves a positive amount, when the interest rate is zero. We shall be interested in the net trades taking place, so it is convenient to consider the derived utility function VE C2(Z, R), Z := 5X: + - {e}, V(zl, z2) := U(e, + zl, e2 + z2). Let us take a look at a given period. Because of the budget constraints, we need only consider the consumption good behavior. The current action as well as the future plan of the young agent is then given by the function z of the following proposition. 1. Given Al and A2, there exists a continuous function, + Z, with the properties:

PROPOSITION

Z: it:+

(i) (ii) (iii) Zlpz,

= is homogeneousof degree 0, : is C’ on B:={(p,p’,,)ElR:+

Ip)8pp’+,),

z(p, pe+,} is the unique element maximizing V on {(z,, z?)~ +p’clzz
where z is typically non-differentiable on the boundary of B contained in lF8: + becauseof the constraint z, =$O. Proof: The result is standard, given the smoothness and boundary assumptions of Al [ 11, Sect. 2.71 and the fact that the income of the agent is a smooth and strictly positive function of prices. 1

Call the coordinate functions z1 and -r2, z,,: RZ, + -+ R, + - {e,}, h = 1,2. From the homogeneity, we might also consider the function z*: R + + -+ Z, defined by z*(P) :=z(fY, 1). We therefore have z(p, ~‘,~)=z*(p/p’,,). From the above, z* is C ’ on )8; so(. Equivalently to the above, let the coordinate functions be zz : IR + + -+ IR + + - {e,,). Being a little imprecise, we shall sometimes refer to the relative intertemporal price 8 as “the interest rate.” Including the time index we have 97 := p,/pT+ 1. Our assumptions so far, however, do not exclude the case of nonmonotonic demand, i.e., it is possible that the offer curve “bends backwards,” such that there may be more than one expected interest rate inducing the young agent to supply a certain quantity of the good. In that case it is impossible to describe the dynamics of the economy by a globally continuous map. We therefore take the unpleasant step and assume gross substitution throughout the rest of this paper:

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Assumption A3. (0: e) is such that D, zi(p, p + , ) < 0 for all (P, P+,)EB. The homogeneity and the budget constraint give us that this is equivalent to having D,z, s-0, D,;, >O, and D,z, ~0 on B, and that Dz:O on )A; CQ(. Now consider the old agent. The behavior is given by the excess demand on the consumption market which is simply m/p, where m is the quantity of money carried over from youth. We now have a full description of the economy, given by the consumer, &, and a money stock, M. DEFINITION. An OLG economy, E, is a pair (&‘, M), where d = (U, e), such that eE rW: + and Cal satisfies Assumptions A.1, A.2, and A3, and

M>O.

To generate temporary equilibria, we shall let agents form price expectations by using a fixed differentiable expectation function, $@3W++, lJ8+ + ). It has as its first argument the current price, and as its second argument the price of the previous period. We shall consider a special class of expectation functions which are very convenient. Definition.

An expectation

function

OG~~(P~P-,)
(i/ is said to be inelastic

where

if

~~(P,P~~):=PD,~~/(P,P~,)/

Note that the set of inelastic expectation functions includes functions taking some arithmetic or geometric average of the current and the lagged price. We are now ready to define a temporary equilibrium price in the obvious way. Definition. For an OLG economy E,p is a temporary equilibrium price given the expectation function II/ and a price history p _, iff --‘I(P,

$(P, p-l))

= M/P.

(TE)

We call the left-hand side of (TE) the supply of the young agent, and the right-hand side the demand of the old agent. By Walras’ law, the above equation implies a clearing also on the money market. The purpose of this paper is to investigate the efficiency properties of sequences of temporary equilibria. We shall call such a sequence a scenario. Definition. Let $ be an expectation function. A scenario for an OLG economy E with the expectation function $ is a sequence {p,,, p,, pz, ...I. where p, is a temporary equilibrium price given $ and the price history

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MICHAEL TEIT NIELSEN

PI&l,

t= 1,2, ... . A stationary scenario is a scenario stationary temporary equilibrium price, t = 1, 2, ... .

where p, = ji, a

It is easily seen that for an OLG economy, there is a unique stationary temporary equilibrium price, given by ji = -M/z,*( 1).

3. RATIONAL

EXPECTATIONS

We shall consider the special case where agents have rational expectations, which in these non-stochastic economies means perfect foresight. DEFINITION. A rational expectations (RE)-scenario for the OLG economy E is a scenario {pO, pl, p?, ... ) for E with the rational expectations function given by HP.

P-l)=

Plc~:+WPn

(for p > M/e ,; clearly, p < M/e, is impossible satisfies the perfect foresight condition Pr+1=II/(P,,

PI-l)>

t

in TE), where the scenario = 1, 2, . .

The rational expectations function, by construction, gives for each current price an expectation such that the market clears, and therefore the temporary equilibrium price is indeterminate. However, we are only interested in scenarios with perfect foresight. For each initial price level ple(p; oo(, an RE-scenario {p,,, pl, pz, ...} then gives us the unique Walrasian equilibrium corresponding to p1 [6]. From the analysis of Gale [6, Sect. 31 we have that if the initial price level is “too low,” i.e., 0 < p1 < p, then the price level will be falling, and in a finite number of periods the real value of the money stock will have increased beyond the endowment of the young agent. Therefore, there are only RE scenarios with p1 2 ~7. If p1 = p, we clearly have pI =p for t = 1,2, ... . On the other hand, if p1 > p, the net trades converge to (0, 0), and the price level is strictly increasing as the interest rate converges close to monotonically to 8. Hence, the economy comes arbitrarily autarchy with a constant inflation rate. We sum up: THEOREM 1. In a rational expectations scenariofor the OLG-economy E, { p,,, pI , p2, . }, where p, > p, we haoe:

(i)

(ii)

Pi+ 1> P,, t = 1, 2, .... B<@ [+ 1< 0, < 1, t = 1, 2, .... and 8, + 8 for t 4 CO,

where 0, := p,/p,+, .

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4. INELASTIC EXPECTATIONS We now consider IE-scenarios, i.e., scenarios where the expectation function is inelastic. Again we utilize A3, this time to get uniqueness of temporary equilibrium and to get global stability. THEOREM 2. For the OLG economy E and the inelastic expectation function $, there exists for each price history p- 1 a unique temporar) equilibrium price p given p-, and $. Furthermore, in an IE scenario .(PO,PI 3PZ?.‘f > for the OLG economy E, M’ehave

PI -+ P

monotonically, for t + K.

Proof. If p increases, from inelasticity I+Qp, p ~ 1) will increase less than proportionally, so 8’ = p/$(p, p _ ]) is increasing in p. As supply increases with 8’ (A3), supply is an increasing function of p. Demand is strictly decreasing, and it converges to zero for p going to infinity, and goes to infinity for p going to zero. For any pr _ 1 > ~7, we have for the unique temporary equilibrium price of period t that ~7< p, < ptp r. For we have that tj(p,- 1, pI- ,) = p,- 1, so supply is the stationary quantity M/p, if p = pr- ,, while demand is M/p,-, < M/p. The inequalities then follow from the properties of the demand and supply functions, as supply is less than IV/J? for p = p. Similarly, for pl- 1< p, we have pip 1< pI < p. 1

(A result which is qualitatively

similar is found’in

5. RELATIVE

[ 131.)

EFFICIENCY

Let us consider the relative efficiency of different types of scenarios. A strong argument for rational expectations is the obvious fact that ex post utility for an agent is maximal when the agent predicts correctly. No agent therefore wishes to be a sloppy predictor. This is the rationalization (!) of agents having rational expectations (of course totally disregarding the complexity or the costs of comprehending the working of an entire economy). It it tempting to infer from the above that “the economy as a whole is better off with rational expectations.” If this were true, there would be a welfare argument for rational expectations, just as there is a welfare argument for Walrasian equilibrium in the Arrow-Debreu economy [4]. However, when performing general equilibrium analysis, we certainly cannot let the constraints be exogenous. Indeed, in our OLG-economy, there is a strong interaction between price expectations and actual prices and thus the actual constraints.

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DEFINITION. The ex post trades in a scenario { pO, p, , pz, . .. } for the OLG economy E with the expectation function I,$ are

(=l(P,> &P,? PI-I)),

WP,,,),

t = 1, 2, ...

For the OLG economy E, the scenario { pO, p,, pz, ...} for E with the expectation function $ is said to Pareto dominate the scenario {pb, p’, , pi, ...I. with pi = p, , for E with the expectation function $‘, if the ex post trades are preferred, i.e., C,(Pr,

NP,, 2 vzl(P:?

PI-- I)), WP,,,) V(Pk

P:-l))?

M/P;+,)

for t = 1, 2, ... and with “ >” for at least one t. Note that the price level in the first period is the same in the two scenarios compared, such that the initial old agent receives the same consumption in the two scenarios. THEOREM 3. Let E be an OLG economy. Let (p,,, p,, pz, ...> be a scenario for E with the inelastic expectation function $J, and let { pb, pi, pi, ...} be the rational expectations scenario for E, with p’, = p,. Then (pO, p, , p2, ... } Pareto dominates (pb, p’, , p;, ... }.

We remark that the difficulty in appraising the relative efficiency of the ex post trades in the IE scenario is that the trades preferred to the ex post trades are typically supported by neither the expected nor the realized budget hyperplane. However, we’ll prove the theorem by showing first that the ex post trades of the IE scenario are preferred to the ex ante trades of this scenario, as there is less inflation than expected. And then in step two, we show that the ex ante trades of the IE scenario are preferred to the ex ante/ex post trades of the RE scenario. Proof: We have that as p, > ji, 0; < 1 < 8,, t = 1,2, .... because pI + , < pr < $(P,, P,- I) < P,-, Therefore, V(=fYW), M/P,+ 1) > v(-:(e)),

I:), as z:(0;) = -e;z:(e;) = f3;M/p, < e,M/p, = M/p,+, , and V is strictly monotonic. Second, by the principle of revealed preferences, we show that the IE ex ante trades are as least as good as the RE ex ante/ex post trades me:)

+ $(P,, P,- 1) a)

a

for

t=

1,

2, ....

which is true iff 0; 2 0; for t = 1, 2, ... . As pr is decreasing monotonically, M/p, is increasing monotonically, and by (TE) and monotonicity of zl, 0: is increasing

monotonically

to 1. On the other hand, & is decreasing

DOMINATION

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FIGURE

monotonically to 8. Therefore -z:(t!)y)=Mlp,, -z:(0;)=Mlp’,, tiveness of 2: that 0; = 0;. 1

EXPECTATIONS

329

1

it s&ices to show 0; b (7,. But as and p;=p,, we have by the injec-

By considering Fig. 1, we see that the proof is driven by the idea that it is necessary and sufficient for the IE scenario to Pareto dominate the RE scenario that 0; Q 0; < 8, for t = 1,2, . Therefore, it is not necessary that there is convergence to the stationary temporary equilibrium in the IE scenario. That is by making expected values exogenous it is possible to construct an allocation generated by “irrational” expected values and by prices that are temporary equilibrium prices given these expected values, which Pareto dominates the corresponding RE scenario. And this is true even if the former has inflation. Another question is whether it is possible to construct, as we did above, such inflationary temporary equilibrium scenarios having endogenousexpected values, but of course exogenous expectations functions. One can show that to have inflation and convergence to autarchy it is necessary that the aggregate excess demand curve has a positive slope. But this implies that the temporary market equilibrium mechanism will be intraperiodically unstable. Furthermore, one can find cases where such an inflationary temporary equilibrium allocation is not Pareto dominating. 6. ABSOLUTE EFFICIENCY

The preceding, somewhat surprising, result could of course not be true if rational expectations scenarios were always Pareto optimal. It is known that Walrasian equilibria in an overlapping generations economy need not be Pareto optimal [l, 21. What is crucial is the curvature of the indifference surfaces and the limit behavior of prices. With our assumptions, we have that the level surfaces of V through the ex post trades all have a Gaussian curvature which is strictly positive and bounded, as the ex post

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trades converge monotonically to (0,O). Furthermore, pi diverges, and in the limit it diverges geometrically, so C: = i l/p, has an upper limit for t going to infinity, and the allocation is suboptimal [ 1, Proposition 5.63. Let us now look at the efficiency properties of different allocations characterized by convergence of the transfer of goods between generations. DEFINITION. A transfer plan for the OLG economy F is a sequence zr, where z0 E R, I?, E R* for t = 1, 2, ... are such that

(i) zO> -e, and Z, > -e for t = 1, 2, .... (ii) z,+z; =o, z;+z;+,=o, t= 1,2, .. . A transfer plan is Pareto optimal iff there exists no improving sequence h,, h, E R, h, E R2 for t = 1, 2, .... such that (z, + h,) is a transfer plan where z()+ ho 2 zo and V(Z, + h,) > V(Z,) for t = 1, 2, .... with “ >” for at least one t. (In the preceding, z: is what is transferred to the agent born just before period t, in period t when the agent is young, while zf is what is transferred to this agent, when the agent is old, in period t + 1. Furthermore, in the definition, (i) gives the condition that the final consumptions lie in the consumption sets, while (ii) gives the endowment constraints.) From the definition and the strict monotonicity of V we see that an improving sequence must be of the form ho = ei, h, = ( -et, E,+ , ), E, > 0, for all t. THEOREM 4. A transfer plan for the OLG economy E in which -7, converges to ZEZ is Pareto optimal if F:= D, V(Z)/Dz V(Z) > 1, and is not Pareto optimal if f < 1.

Proof First of all, z, belongs to a compact set, so the Gaussian curvature is bounded to lie in some strictly positive interval, (a; b). Similarly the marginal rate of substitution, rr := D, V(z,)/D, V(z,), is bounded. Second, for an improving sequence, we must have that E,+ i > r,&, > 0 for all t>, T when s7>0. Let r be less than one. By convergence of zI, we have convergence of rr, as V is C2. Let r’ := sup rtr and let T be such that r, T (such a T exists, as r, converges to r less than one). A sufficient condition for a sequence to be improving is that (-et, E,, i) lie inside the circles with centers (m,/(2bdw), 1/(264x)), where ml= r’ for t Q T, and m, = k for t > T, and with radius 1/2b. Let ( - x, J-‘), x, y > 0, lie inside both those circles, and let d := y/x. We then have improving sequence by E, = min(x, dpT.u) and E,=~‘~‘E~ for t=2, 3, .... r, and E,=E= for t> T. Conversely, let r be larger than one. Then there is no improving sequence, as there exists a c > 0 and a T> 0, such that r, > (1 + c) for t > T,

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so for a potential improving sequence we have E, > (1 + c)‘- 7‘x for some strictly positive X, such that (z: - E,) < ( -e, ) for t large, i.e., the suggested improvement breaks the endowment constraints. To complete, if ? is one, the transfer plan is Pareto optimal, if n:=, ri does not converge to zero for t going to infinity. For in that case, for every suggested improving sequence, E, is bounded below by a strictly positive number. Also, as r, converges to one, E,+ , /et has limit inferior at least one. In fact, because of the endowment constraints, giving an upper bound on E,, we must have that E,, 1 /E, converges to one. As the indifference curve through 2 has strictly positive curvature, the suggested sequence is not improving for generations in the distant future; a contradiction. This is obviously the case if r, converges to 1 from above. However, it is possible to construct transfer plans where r, converges slowly to 1 from below, such that I-I:=, r, converges to 0 for t going to infinity, i.e., the transfer plan is suboptimal. 1 THEOREM 5. Let {pO, p,, pZ, ... 1 be a scenario for the OLG economy E with the inelastic expectation function $. Then the ex post trades of the inelastic expectations scenario constitute a Pareto optimal transfer plan.

Proof: We know that the ex post trades converge to the stationary “golden rule” values. From the remarks in the proof of Theorem 4, we have to make sure that the marginal rates of substitution converge sufficiently fast to one. Let us analyze the limit behavior of the ex post trades and the marginal rates of substitution. Let dp, := p, - p, dz,(t)

:==:(P,/$(P,,

d=(t)=

(dz,(t),

d-7>(t) := M/p,+,

p,-,))--‘:(l), d;,(t)),

-=(=,(I),

-M/p,

dr,=r,-

M/F),

1.

By differentiation of the temporary market equilibrium equation at (p, p), we get the following expressions, where the S-terms are defined residually and have convenient convergence properties [9] : dp,+dpr-, 4

1 +k =7+6Jt-

l),

lim 6,(t) = 0, I -

r+1

,x

where k:=Dz:(l-q)/(Dz:(I--)-M/p) and O
-q)(l

+k))/Mk*,

lim s,(t)=O.

r--1 r

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MICHAEL TEIT NIELSEN

Differentiation of dr, = (D, V(Z+ dz(t)) - D, V(5+ dz(t)))/D, V(Z+ dz(t)) allows us to write lim 6,(f) = 0. I+ %

dr,/dr, - , = k + 6,(t),

In the limit, the marginal rate of substitution is converging geometrically fast to 1, so there exists a T and a k”, 0
ldr,l
for t> T. Let g := CT:,’ f,

Now, n:=,

Idr,l
Idr,l/(l

t>T Idrll, then -1)~

aci,

r, > 0 iff C;?&, In r, > --co, but we can write

lnr,=r,-1

+b,(t)=dr,+b,(t),

where lim, _ ~ Ii?,(t)l/ldr,l =O. But this means there exists a T’ such that

for t > T’, Id,(t)1 < Idr,l, so Cj’L, Id,(t)1 G CT:, Id,(t)l + C,“= , Id-J, and as we saw, x:,“=r Idr,l is bounded C;“=, dr,+C,“=,

~,W>

--Go.

so Cr=“=, 16,(t)/ < 00. Thus, C,“=, In r,= I

It may seem puzzling that we have Pareto optimality in a scenario where agents do not predict correctly. However, note that in the unidimensional case (only one consumption good per period, and only one agent per generation), any allocation can be supported as a market equilibrium, using marginal rates of substitution to construct supporting prices. In fact, it is exactly this idea which lies behind the proof of Theorem 5. In the multi-dimensional case, an allocation, even if it is generated by temporary equilibrium prices, will only be market-supportable by a fluke, and we would generally expect suboptimality. 7. CONCLUDING

REMARKS

We conclude that when agents form their expectations using an “irrational” inelastic expectation function, not only do we get convergence of trades to “golden rule” values, but every generation is better off than in the corresponding rational expectations scenario, and in fact the allocation generated is optimal, in spite of systematic prediction errors. One might wonder about the economic explanation behind the result, which may seem paradoxical at first sight-after all, etymologically, “paradox” means “contradicting what was expected” ! The explanation is that there is an

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interaction between prices and price expectations [S, 111. When agents have inelastic expectations, the economy is kept away from the unstable path where the gains from trade are being eroded because of inflation, instead being led towards the golden rule equilibrium. A disturbing fact is that this is, in a sense, a non-robust result, for each agent would be better off, individually, if predicting correctly, given that prices are what the are; and therefore each agent has an incentive to change his or her way of forming expectations. It may be argued that the domination result is obtained in a very simplistic model with very restrictive assumptions, but this paper is thought of as a kind of counterexample, and therefore existence, and not generality, has been stressed. Let us comment briefly on the multi-dimensional case: The instability of the stationary temporary equilibrium which is necessary for the suboptimality of a rational expectations scenario is not always found in the multi-dimensional case [lo]. So the domination result certainly does not hold true in general in the multi-good case, but we might conjecture that robust examples can be constructed. We also saw that in the uni-dimensional case, the inelastic expectations scenario was Pareto optimal, even if agents predicted incorrectly. This result, on the other hand, seems to be an artefact of the one-good model, as briefly indicated in Section 6. In the uni-dimensional case, agents always have the same marginal rate of substitution inside a period, so we can construct supporting market equilibrium prices, and furthermore, those artificial prices converge sufficiently fast because of the (temporary equilibrium) market mechanism. However, in the multi-dimensional case, it seems improbable that agents, when not predicting the correct prices, will have the same intraperiod marginal rates of substitution, unless preferences are separable between periods. Finally, a remark on the policy implications: Bohm and Pahukka [3] show Pareto domination of quantity-constrained equilibria over Walrasian equilibria. It might be said that the Bohm and Puhakka paper gives an argument for government intervention: An omniscient planner could dictate quantity constraints and thereby make everybody better off. Such a conclusion is not easily derived from the present paper, as it is hard to see how a central authority could monitor, or even govern, the subjective expectations of individual agents.

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AND M. PUHAKKA, Rationing and optimality in overlapping generations models, working paper, University of Mannheim/University of Pennsylvania, January 1985. 4. G. DEBREU, “Theory of Value,” Cowles Foundation, New Haven, CT, 1959. 5. G. FUCHS AND G. LAROQUE, Dynamics of temporary equilibria and expectations. Econometrica 44 (1976). 1157-l 178. 6. D. GALE, Pure exchange equilibria of dynamic economic models, J. Econ. Theory 6 3. V. BOHM

(1973).

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I. J.-M. GRANDMONT, “Money and Value,” Cambridge Univ. Press, London/New York, 1983. 8. J.-M. GRANDMONT, On endogenous competitive business cycles, Econometrica 53 (1985), 995-1045. 9. M. W. HIRSCH AND S. SMALE, “Differential Equations, Dynamical Systems, and Linear Algebra,” Academic Press, New York, 1974. 10. T. J. KEHOE AND D. K. LEVINE, Regularity in overlapping generations exchange economies, 1. Math. Econ. 13 (1984), 69-93. 11. A. MAS-COLELL, “The Theory of General Equilibrium. A Differentiable Approach,” Cambridge Univ. Press, London/New York, 1985. 12. P. A. SAMUELSON,An exact consumption-loan model of interest with or without the social contrivance of money, J. Polit Econ. 66 (1958), 467482. 13. G. TILLMANN, Stability in a simple pure consumption loan model, J. Econ. Theor?, 30 ( 1983), 3 15-329.